(M-5) Deriving Approximate Results

A Preliminary Derivation

Given a fraction a/b, one may multiply or divide its top and bottom ("numerator and denominator") by the same number c:

(a/b) = (ac)/(bc)

where (remember?) the two letters ac stand for "a
times c" and similarly for bc.

That is so because (c/c) = 1, no matter what the value of c is (except of course zero: "Thou Shalt not Divide by Zero") and multiplying anything by 1 does not change its value. In multiplying fractions, the rule is to multiply top with top, bottom with bottom, so we get

(a/b) (c/c) = (ac)/(bc)

As for dividing top and bottom by the same number d

(a/b) = [a/d]/ [b/d]

it follows at once from the preceding, if we choose the number c to equal 1/d.

Working with Small Quantities

Some equations, identities or formulas contain small quantities, and these can be made much simpler and easier to use by sacrificing a little accuracy. In fact, some equations which have no simple solution at all (like Kepler's equation in section (12a)) can yield in this way an approximate solution, often good enough for most uses, or else open to further improvement.

Many such calculations make use of the following observation. When we derive squares, 3rd powers, 4th powers etc. of numbers larger than 1, the results are always bigger, while for numbers smaller than 1, the results are always smaller. For example:

power

More than 1

Less than 1

number

10

0.1

square

100

0.01

3rd power

1000

0.001

4th power

10,000

0.0001

The above also holds for negative numbers, if one understands "bigger" and "smaller" to refer to the absolute value (the value without sign). For instance:

power

More than 1

Less than 1

number

– 10

– 0.1

square

100

0.01

3rd power

– 1000

– 0.001

4th power

10,000

0.0001

5th power

– 100,000

– 0.00001

Say z is a number much smaller than 1 (written z << 1, or for absolute values |z| << 1). Then by the identity of section M-4

(1 – z)(1 + z) = 1 – z2

Since z2 is much smaller than 1 or z, we can write, using the symbol ~ for "approximately equal"

(1 – z)(1 + z) ~ 1

and dividing both sides by (1 – z)

(1+z) ~ 1/(1– z)

(Many texts use the symbol ~ not alone but placed above an equal sign; however, that combination is not available for web documents). For example (check with your calculator)

If z = 0.01, (1+z) = 1.01, (1– z) = 0.99,

then 1/(1– z) = 1/0.99 = 1.010101...

which is close enough to (1+z) for many purposes.

The basic rule is: one may neglect small quantities like z, z2, z3 etc. when they are added to (or subtracted from) something much bigger. One may not do so if they are just multiplied or divided, because then, if they are removed, nothing is left of the expression containing them.

Here z can be either positive or negative. If we write z = – y, where y is a small number of opposite sign, we get

(1– y) ~ 1/(1+y)

which is another useful result, valid for any small number. If that small number is again renamed and is now called z (not the same z as before, of course), we get

(1– z) ~ 1/(1+z)

which can also be obtained from the earlier equation

(1 – z)(1 + z) ~ 1

by dividing both sides by (1 + z).

In section (34a) where the distance to the Lagrangian point L1 is derived, it turns out necessary to approximate 1/[1– z]3. You start from (1+z) ~ 1/(1– z) and raise both sides to their 3rd powers:

(1+z)3 ~ 1/(1– z)3

Multiply out the left side:

(1 + z)3 = (1+z)(1+z)(1+z) = (1 + 3z + 3z2 + z3)

However, if z2 and z3 are much smaller than z, then dropping the terms containing them only increases the error slightly, leaving

Note: Why not divide by zero? It does not work. There exists no number like 1/0 (except maybe infinity, which is not a regular number), and use of expressions like 0/0 can lead to contradictions such as 2 = 3.